Sufficient condition for asymptotic-$\ell_{p}$ in terms of spreading models?

Question

Let $(X,\|\cdot\|)$ be a Banach space with a Schauder basis and fix $p\in[1,\infty]$. Suppose that $X$ is asymptotic-$\ell_{p}$ with respect to this basis. It is known that the closed linear span of every (nontrivial) spreading model of $X$ is isomorphic to $\ell_{p}$ if $X$ is reflexive and at least contains an isomorphic copy of $\ell_{p}$ in general (replace $\ell_{p}$ by $c_{0}$ if $p=\infty$). In other words, the global asymptotic geometry of $X$ gives some information about the local asymptotic geometry.

Do there exist any known converse results? For example, are there general hypotheses that, in combination with the closed linear span of every spreading model containing an isomorphic copy of $\ell_{p}$, ensure that $X$ itself will be asymptotic-$\ell_{p}$?

Note that this question is cross-listed here on MSE.

Answer 1

The answer to the question you formulated is no in a very strong sense. For all $1<p<\infty$ there exists a reflexive space $X$ with an unconditional basis so that $X$ for all $\varepsilon>0$ every normalized weakly null sequence in $X$ admits a subsequence $1+\varepsilon$-equivalent to the unit vector basis of $\ell_p$ (so all spreading models are isomorphic to $\ell_p$) and yet its asymptotic structure contains $\ell_q^n$'s for some $p\neq q$. See Example 4.2 of this paper of Odell and Schlumprecht

There are many variation of the problem relating different asymptotic structures, and often the right question is to ask for a subspace with a better asymptotic structure under the assumption of, say, all spreading models being isomorphic to some $\ell_p$. The remarkable paper of Argyros and Motakis (that Kevin already referred to) gives some definite answers to some of these difficult questions. See the references therein to discover older results.